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– Model of axisymmetric two-layer cylinder. (Boldfaced lines indicate where boundary conditions are particularly challenging).
Source publication
Closed-form analytical expressions are derived for the displacement field and corresponding stress state in two-layer cylinders subjected to pressure and thermal loading. Solutions are developed both for cylinders which are fully restrained axially (plane strain) and for axially loaded and spring-mounted cylinders, assuming that the combined two-la...
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Citations
... where N app is an applied axial load due to for instance changes in temperature, internal pressure or second order load effects due to lateral displacement of the pipe (typically caused by hydrodynamic loading), k is the axial spring stiffness (at each segment end), u z is the axial displacement at the axial coordinate z e (=L/2, relative to the segment mid-length) at either cylinder end. It has been shown previously by Vedeld and Sollund [30] that the exact displacement field for two-layer cylinders exposed to temperature, internal and external pressure, as well as axial forces, may be written as ...
... Based on full three dimensional elasticity theory and the displacement fields defined by Eq. (7), stress fields have been derived in [30] for both the stainless steel liner (inner layer) and the backing steel (outer layer). The stress field in cylindrical coordinates for the inner layer is given by r rr r hh r zz ...
... For two-layer cylinders, the same Lamé displacement field used for the single layer pipe (Eq. (7)) can be applied successively for each layer [30]. First, a solution will be obtained for a pipe segment subjected to a temperature change DT only. ...
Closed-form analytical expressions are derived for the displacement field and corresponding stress state in two-layer cylinders subjected to pressure and thermal loading. Solutions are developed both for cylinders that are fully restrained axially (plane strain) and for cylinders that are axially loaded and spring-mounted. In the latter case, it is assumed that the combined two-layer cross section remains plane after deformation (generalized plane strain). The analytical solutions are verified by means of detailed three-dimensional finite element (FE) analyses, and they are easily implemented in, and suitable for, engineering applications. The chosen axial boundary conditions are demonstrated to be particularly relevant for pipeline and piping applications. By applying the exact solutions derived in the present study to typical offshore lined or clad pipelines, it is demonstrated that thermal expansion of the liner or clad layer may cause higher tensile hoop stresses in the pipe steel wall than accounted for in current engineering practice. It is shown that repeated cycles of start-up and shut-down phases for lined or clad pipelines cause significant plastic stress cycles in liners or claddings, which may pose a risk to the integrity of such pipelines.
... For a circular cylinder deformed symmetrically with respect to the z-axis and with zero shearing stresses, the following differential equation of equilibrium may be derived for the radial direction (Timoshenko and Goodier, 1951;Vedeld and Sollund, 2013a): ...
... Convergence of the numerical results to 5 significant digits was considered sufficient and element resolution was refined to obtain this accuracy. Further details on the FEA modeling are described by Vedeld and Sollund (2013a;2013b). ...
Two independent sets of analytical solutions have been derived for calculation of displacements and stresses in elastic multi-layer cylinders subjected to both pressure and thermal loading. The solutions are computationally efficient and easily implemented, and thus suitable for engineering applications. In particular, the novel solutions are relevant for offshore pipelines and risers, since the impact of internal and external coating layers on the stress state in the pipe wall often is disregarded or assessed by simplified methods in current pipeline engineering practice. Recursive solutions are obtained both for the case of uniform temperature in each cylinder layer and for the case of radially varying temperature in each cylinder layer. Both plane stress and plane strain conditions are considered. In addition, a recursive solution of the heat equation is derived for steady-state conditions. Applicability of the solutions is illustrated by using them on an offshore pipeline with a corrosion-resistant liner and multi-layered thermal insulation coating. It is demonstrated that coating layers, generally disregarded in conventional pipeline design, may have a non-negligible impact on hoop stresses and the true wall axial force.
... where N app is an applied axial load due to for instance changes in temperature, internal pressure or second order load effects due to lateral displacement of the pipe (typically caused by hydrodynamic loading), k is the axial spring stiffness (at each segment end), u z is the axial displacement at the axial coordinate z e (=L/2, relative to the segment mid-length) at either cylinder end. It has been shown previously by Vedeld and Sollund [30] that the exact displacement field for two-layer cylinders exposed to temperature, internal and external pressure, as well as axial forces, may be written as ...
... Based on full three dimensional elasticity theory and the displacement fields defined by Eq. (7), stress fields have been derived in [30] for both the stainless steel liner (inner layer) and the backing steel (outer layer). The stress field in cylindrical coordinates for the inner layer is given by r rr r hh r zz ...
... For two-layer cylinders, the same Lamé displacement field used for the single layer pipe (Eq. (7)) can be applied successively for each layer [30]. First, a solution will be obtained for a pipe segment subjected to a temperature change DT only. ...
The article presents the solution of the axisymmetric problem of the stress-strain state of a two-layer hollow thick-walled cylinder. A model of limited length with loaded ends is considered. It is assumed that the interaction of layers is implemented through a contact layer. The contact layer is considered as a transversally anisotropic elastic medium with such parameters that it can be represented as a set of short elastic rods that are not connected to each other and are normally oriented to the contact surface. Such an assumption makes possible to obtain an analytical solution of the problem presented in a closed form. The solution obtained allows us to calculate essentially inhomogeneous stress and strain fields, including stress concentrations, and also satisfies all boundary conditions.
A link for the full paper is valid for 50 days and found at:
https://authors.elsevier.com/a/1XDgOW4G4FdRq
Abstract:
To ensure high content temperature, some offshore pipelines have insulation coating made from polypropylene
or polyethylene. Typical coating systems have significantly lower stiffness than ordinary carbon manganese
steel, but also significantly higher temperature expansion coefficients. Due to the differences in material properties,
insulation coating systems have a non-trivial influence on the critical design parameter called the effective
axial force in offshore pipelines. Current offshore design codes do not contain guidance on how to include the
effects of coating systems on the effective axial force. In this paper, an exact analytical formula for the effective
axial force in pipelines with arbitrarily many layers is deduced, accounting also for temperature variation along
the radial coordinate. A simplified approximate formula, which is more suitable for use in engineering contexts,
is developed and its validity verified by comparisons to results of the exact one.
